A general theorem on inverse semigroups with finiteness conditions (Q1091478)

Let S be an inverse semigroup with zero and \(\kappa\) be an abstract condition on inverse semigroups. In this paper the author shows the following: 1. Suppose that (i) every inverse subsemigroup of an inverse \(\kappa\)-semigroup has the property \(\kappa\) ; (ii) every basis of an inverse \(\kappa\)-semigroup is finite. If S is an inverse \(\kappa\)- semigroup, then (*) the set of all idempotents of S is finite and every maximal subgroup of S has the property \(\kappa\). 2. Suppose that a finiteness condition \(\kappa\) satisfies: (iii) every inverse semigroup which is a finite union of inverse \(\kappa\)-semigroups has the property \(\kappa\) ; (iv) every ideal extension of an inverse \(\kappa\)-semigroup by an inverse \(\kappa\)-semigroup has the property \(\kappa\) ; (v) every Brandt semigroup B(G,2) where G is a \(\kappa\)-group has the property \(\kappa\). If an inverse semigroup S satisfies (*), then S is a \(\kappa\)- semigroup.